{
  "compose": 1,
  "id": "partee-triangle",
  "title": "Partee 1986 — The Type-Shifting Triangle",
  "notation": "cc",
  "domain": {
    "multiLetterNames": true,
    "constants": {
      "e": "j m h p"
    },
    "variables": {
      "e": "x y z",
      "et": "P Q X Y",
      "<<e,t>,t>": "T U"
    }
  },
  "lexicon": [
    {
      "words": [
        "John"
      ],
      "denotation": "j"
    },
    {
      "words": [
        "Mary"
      ],
      "denotation": "m"
    },
    {
      "words": [
        "Hesperus"
      ],
      "denotation": "h"
    },
    {
      "words": [
        "Phosphorus"
      ],
      "denotation": "p"
    },
    {
      "words": [
        "man"
      ],
      "denotation": "Lx.man(x)"
    },
    {
      "words": [
        "king"
      ],
      "denotation": "Lx.king(x)"
    },
    {
      "words": [
        "president"
      ],
      "denotation": "Lx.president(x)"
    },
    {
      "words": [
        "teacher"
      ],
      "denotation": "Lx.teacher(x)"
    },
    {
      "words": [
        "student"
      ],
      "denotation": "Lx.student(x)"
    },
    {
      "words": [
        "tall"
      ],
      "denotation": "Lx.tall(x)"
    },
    {
      "words": [
        "walks",
        "walk"
      ],
      "denotation": "Lx.walk(x)"
    },
    {
      "words": [
        "arrived",
        "arrive"
      ],
      "denotation": "Lx.arrive(x)"
    },
    {
      "words": [
        "is",
        "are"
      ],
      "denotation": "LP.P"
    },
    {
      "words": [
        "every"
      ],
      "denotation": "LX.LY.Ax[X(x) -> Y(x)]"
    },
    {
      "words": [
        "a",
        "an"
      ],
      "denotation": "LX.LY.Ex[X(x) & Y(x)]"
    },
    {
      "words": [
        "THE"
      ],
      "displayAs": "the",
      "denotation": "LX.LY.Ex[Ay[X(y) ↔ y = x] & Y(x)]"
    },
    {
      "words": [
        "the"
      ],
      "denotation": "LX.Ix.X(x)"
    },
    {
      "words": [
        "andT"
      ],
      "displayAs": "and",
      "denotation": "LU.LT.LP.[T(P) & U(P)]"
    },
    {
      "words": [
        "andP"
      ],
      "displayAs": "and",
      "denotation": "LQ.LP.Lx.[P(x) & Q(x)]"
    }
  ],
  "rules": {
    "composition": {
      "functionApplication": true,
      "predicateModification": true,
      "nonBranchingNodes": true
    },
    "typeShifts": [
      "lift",
      "lower",
      "ident",
      "iota",
      "aop",
      "be"
    ]
  },
  "exercises": [
    {
      "id": "g1",
      "title": "A. Three types for NPs",
      "instructions": "Referential, predicative, quantificational: John as e, a fool as ⟨e,t⟩, every man as ⟨⟨e,t⟩,t⟩ (Partee 1986, §2).",
      "derivations": [
        {
          "id": "d1",
          "tree": "[.t [.DP John ] [.IV walks ] ]",
          "sentence": "John walks.",
          "targets": [
            "walk(j)"
          ],
          "reading": {
            "section": "1"
          }
        },
        {
          "id": "d2",
          "tree": "[.t [.DP Mary ] [.Pred [.Cop is ] [.AP tall ] ] ]",
          "sentence": "Mary is tall.",
          "targets": [
            "tall(m)"
          ],
          "reading": {
            "section": "1"
          }
        },
        {
          "id": "d3",
          "tree": "[.t [.DP [.Det every ] [.CN man ] ] [.IV walks ] ]",
          "sentence": "Every man walks.",
          "targets": [
            "Ax[man(x) -> walk(x)]"
          ],
          "reading": {
            "section": "1"
          }
        }
      ]
    },
    {
      "id": "g2",
      "title": "B. BE — predicative NPs (defs. 14–16)",
      "instructions": "English be means apply-predicate, λP.λx.P(x) (Williams 1983; Partee’s (16)). When its complement is a quantifier, the BE shift lowers it to ⟨e,t⟩: BE(P) = λx.P(λy.y = x).",
      "derivations": [
        {
          "id": "d1",
          "tree": "[.t [.DP John ] [.Pred [.Cop is ] [.DP [.Det a ] [.CN man ] ] ] ]",
          "sentence": "John is a man.",
          "hints": [
            "a man is a quantifier λY.∃x[man(x) ∧ Y(x)] — but is wants a predicate. Apply the BE shift.",
            "BE(⟦a man⟧) = λx.∃y[man(y) ∧ y = x] = λx.man(x) — Partee’s computation (21): BE(a man) = man."
          ],
          "targets": [
            "man(j)"
          ],
          "reading": {
            "section": "2"
          }
        },
        {
          "id": "d2",
          "tree": "[.t [.DP John ] [.Pred [.Cop is ] [.DP [.Det THE ] [.CN president ] ] ] ]",
          "sentence": "John is the president.",
          "targets": [
            "Ez[Ay[president(y) ↔ y = z] & z = j]"
          ],
          "reading": {
            "section": "2"
          }
        }
      ]
    },
    {
      "id": "g3",
      "title": "C. THE and iota (def. 14, Diagram 2)",
      "instructions": "THE is the generalized-quantifier definite; iota is its e-type cousin. When uniqueness holds, lift(iota(king)) = THE(king).",
      "derivations": [
        {
          "id": "d1",
          "tree": "[.t [.DP [.Det the ] [.CN king ] ] [.IV walks ] ]",
          "sentence": "The king walks.",
          "targets": [
            "walk(Ix[king(x)])"
          ],
          "reading": {
            "section": "3"
          }
        },
        {
          "id": "d2",
          "tree": "[.t [.DP [.Det THE ] [.CN king ] ] [.IV walks ] ]",
          "sentence": "The king walks. (GQ route)",
          "targets": [
            "Ex[Ay[king(y) ↔ y = x] & walk(x)]"
          ],
          "reading": {
            "section": "3"
          }
        }
      ]
    },
    {
      "id": "g4",
      "title": "D. ident — equatives",
      "instructions": "ident maps j to λx[x = j]: a name in predicate position.",
      "derivations": [
        {
          "id": "d1",
          "tree": "[.t [.DP Hesperus ] [.Pred [.Cop is ] [.DP Phosphorus ] ] ]",
          "sentence": "Hesperus is Phosphorus.",
          "targets": [
            "h = p"
          ],
          "reading": {
            "section": "4"
          }
        }
      ]
    },
    {
      "id": "g5",
      "title": "E. Coercion by conjunction (§1.2; Partee & Rooth 1983)",
      "instructions": "Unlike-type conjunctions force shifts: the e-type definite LIFTs to conjoin with a quantifier; a predicative AP conjoins with a BE-shifted NP.",
      "derivations": [
        {
          "id": "d1",
          "tree": "[.t [.DP [.DP [.Det the ] [.CN teacher ] ] [.ConjP [.Conj andT ] [.DP [.Det every ] [.CN student ] ] ] ] [.IV arrived ] ]",
          "sentence": "The teacher and every student arrived.",
          "targets": [
            "arrive(Ix[teacher(x)]) & Ax[student(x) -> arrive(x)]"
          ],
          "reading": {
            "section": "5"
          }
        },
        {
          "id": "d2",
          "tree": "[.t [.DP Mary ] [.Pred [.Cop is ] [.PredP [.AP tall ] [.ConjP [.Conj andP ] [.DP [.Det a ] [.CN teacher ] ] ] ] ] ]",
          "sentence": "Mary is tall and a teacher.",
          "targets": [
            "tall(m) & teacher(m)"
          ],
          "reading": {
            "section": "5"
          }
        }
      ]
    }
  ],
  "reading": {
    "format": "latex",
    "markdown": "# Partee 1986 · The Type-Shifting Triangle\n\nCompanion to Barbara Partee, \"Noun Phrase Interpretation and Type-Shifting\nPrinciples\" (1986). Definitions and computations below are Partee's own,\ncited by her numbering; this reading was checked against Partee's own\nsummary of the paper (Partee 2006, Lecture 14).\n\n## 1 Three types for NPs\n\nMontague gave every NP the quantifier type ⟨⟨e,t⟩,t⟩. Partee's resolution,\nas she summarizes it (2006, Lecture 14): every NP has a ⟨⟨e,t⟩,t⟩ meaning,\nand some also have e and/or ⟨e,t⟩ meanings, related by general shifting\nprinciples. Her evidence: e-type NPs\nlicense discourse anaphora (her (9)–(10): *John / the man / a man walked in.\nHe looked tired* vs. *every man / no man … \\*He looked tired*), and\npredicative positions select ⟨e,t⟩ NPs (her (11)–(12): *Mary considers John\ncompetent in semantics and an authority on unicorns*).\n\n\\pex\n\\a John walks. (e)\n\\a Mary is tall. (⟨e,t⟩ predicate)\n\\a Every man walks. (⟨⟨e,t⟩,t⟩)\n\\xe\n\n## 2 BE — predicative noun phrases\n\nPartee's (15)–(16), following Williams: English *be* is just apply-predicate,\nλP.λx.P(x). When its complement is a quantifier, the **BE** shift applies\n(her (14)): BE(𝒫) = λx.𝒫(λy.y = x) — find the singletons in the quantifier,\ncollect their members. Her computations (21):\n\n\\begin{derivation}\nBE(a man)     = λx.man(x)\nBE(John)      = λx.x = j\nBE(no man)    = λx.¬man(x)\nBE(every man) = λx.Ay[man(y) -> y = x]\n\\end{derivation}\n\nBE is a Boolean homomorphism, and the unique one making her Diagram 3\ncommute (Facts (19)–(20)) — that is the \"naturalness\" argument, and A is\nnatural in part as an inverse of BE: BE(A(P)) = P.\n\n\\ex John is a man.\n\\xe\n\n\\ex John is the president.\n\\xe\n\nFor the second: BE(THE(president)) applied to j. The worksheet's target is\nthe form the shift derives, ∃z[∀y[president(y) ↔ y = z] ∧ z = j]; Partee's\n(17) states the equivalent λx[president(x) ∧ ∀y[president(y) ↔ y = x]].\nHer (18) records the bare-nominal contrast: *John is {the president /\npresident}* but *John is {the teacher / \\*teacher}*.\n\n## 3 THE and iota\n\nTHE (her (14)): Q ⇒ λP[∃x[∀y[Q(y) ↔ y = x]] ∧ P(x)] — the generalized\nquantifier definite. iota: P ⇒ ιx.P(x), partial. When uniqueness holds the\ntriangle commutes: lift(iota(king)) = THE(king) and lower(THE(king)) =\niota(king). Derive *the king walks* both ways and compare.\n\n## 4 ident — equatives\n\nident (Diagram 1): j ⇒ λx[x = j], a name in predicate position. The laws of\nthe triangle: lower(lift(j)) = j and iota(ident(j)) = j — lift/lower and\nident/iota are inverse pairs (up to partiality).\n\n\\ex Hesperus is Phosphorus.\n\\xe\n\n## 5 Coercion by conjunction\n\nPartee's own motivation for the whole system (§1.2) comes from Partee &\nRooth 1983: unlike-type conjunctions force shifts. The e-type definite lifts\nto conjoin with a quantifier; a predicative AP conjoins with a BE-shifted\nindefinite. (The full conjunction story has its own worksheet on this page.)\n\n\\pex\n\\a The teacher and every student arrived.\n\\a Mary is tall and a teacher.\n\\xe\n\n## Credits\n\nPartee, B. H. (1986). Noun phrase interpretation and type-shifting\nprinciples. In Groenendijk, de Jongh & Stokhof (eds.), *Studies in Discourse\nRepresentation Theory and the Theory of Generalized Quantifiers*, 115–143.\nForis. Reprinted in Portner & Partee (2002) and Partee (2004). — Partee,\nB. H. (2006). *The Structure of Meaning*, Lecture 14\n(people.umass.edu/partee) — definitions, data and computations quoted from\nit with Partee's numbering. — Williams, E. (1983), for the analysis of\n*be*. Notes text: quotes and numbered definitions as marked, plus flagged\nrendering notes (the derived-form target in §2); remaining prose is\nsignposting."
  },
  "meta": {
    "author": "Thomas Stephen",
    "license": "MIT",
    "difficulty": "challenge"
  }
}